Abstract
Tight oil/gas medium is a special porous medium, which plays a significant role in oil and gas exploration. This paper is devoted to the derivation of wave equations in such a media, which take a much simpler form compared to the general equations in the poroelasticity theory and can be employed for parameter inversion from seismic data. We start with the fluid and solid motion equations at a pore scale, and deduce the complete Biot’s equations by applying the volume averaging technique. The underlying assumptions are carefully clarified. Moreover, time dependence of the permeability in tight oil/gas media is discussed based on available results from rock physical experiments. Leveraging the Kozeny-Carman equation, time dependence of the porosity is theoretically investigated. We derive the wave equations in tight oil/gas media based on the complete Biot’s equations under some reasonable assumptions on the media. The derived wave equations have the similar form as the diffusive-viscous wave equations. A comparison of the two sets of wave equations reveals explicit relations between the coefficients in diffusive-viscous wave equations and the measurable parameters for the tight oil/gas media. The derived equations are validated by numerical results. Based on the derived equations, reflection and transmission properties for a single tight interlayer are investigated. The numerical results demonstrate that the reflection and transmission of the seismic waves are affected by the thickness and attenuation of the interlayer, which is of great significance for the exploration of oil and gas.
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References
Biot M A. 1956a. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J Acoust Soc Am, 28: 168–178
Biot M A. 1956b. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J Acoust Soc Am, 28: 179–191
Biot M A. 1962. Generalized theory of acoustic propagation in porous dissipative media. J Acoust Soc Am, 34: 1254–1264
Biot M A. 1973. Nonlinear and semilinear rheology of porous solids. J Geophys Res, 78: 4924–4937
Bourbié T, Coussy O, Zinszner B. 1987. Acoustics of Porous Media. Houston: Gulf Publishing Co.
Carman P C. 1961. Lecoulement des GazaTravers les Milieux Poreux, Bibliothéque des Sciences et Techniques Nucleaires. Paris: Presses Universitaires de France
Chen Y, Huang T F, Liu E R. 2009. Rock Physics Handbook (in Chinese). Hefei: University of Science and Technology of China Press
Cheng Y F, Yang D H, Yang H Z. 2002. Biot/squirt model in viscoelastic porous media. Chin Phys Lett, 19: 445–448
Cruz V D L, Sahay P N, Spanos T J T. 1993. Thermodynamics of porous media. Proc R Soc Lond A, 443: 247–255
de la Cruz V, Spanos T J T. 1985. Seismic wave propagation in a porous medium. Geophysics, 50: 1556–1565
de la Cruz V, Spanos T J T. 1989. Seismic boundary conditions for porous media. J Geophys Res, 94: 3025–3029
Diallo M S, Appel E. 2000. Acoustic wave propagation in saturated porous media: Reformulation of the Biot/Squirt flow theory. J Appl Geophys, 44: 313–325
Diallo M S, Prasad M, Appel E. 2003. Comparison between experimental results and theoretical predictions for P-wave velocity and attenuation at ultrasonic frequency. Wave Motion, 37: 1–16
Fan L S, Zhu C. 2005. Principles of Gas-Solid Flows. Cambridge: Cambridge University Press
Fung Y. 1994. A first Course in Continuum Mechanics. 3rd ed. New Jersey: Prentice-Hall, Inc.
Goloshubin G M, Korneev VA. 2000. Seismic low-frequency effects from fluid-saturated reservoir. In: Proceedings SEG Meeting (Calgary), 70th Annual International Meeting, SEG. Expanded Abstracts. 1671–1674
Han W, Gao J, Zhang Y, Xu W. 2020. Well-posedness of the diffusive-viscous wave equation arising in geophysics. J Math Anal Appl, 486: 123914
He Y, Chen T, Gao J. 2019. Unsplit perfectly matched layer absorbing boundary conditions for second-order poroelastic wave equations. Wave Motion, 89: 116–130
He Y, Chen T, Gao J. 2020. Perfectly matched absorbing layer for modelling transient wave propagation in heterogeneous poroelastic media. J Geophys Eng, 17: 18–34
He Z, Xiong X, Bian L. 2008. Numerical simulation of seismic low-frequency shadows and its application. Appl Geophys, 5: 301–306
Hickey C J, Spanos T J T, Cruz V. 1995. Deformation parameters of permeable media. Geophys J Int, 121: 359–370
Johnston D H, Toksöz M N, Timur A. 1979. Attenuation of seismic waves in dry and saturated rocks: II. Mechanisms. Geophysics, 44: 691–711
Kang Y. 2016. Resource potential of tight sand oil & gas and exploration orientation in China (in Chinese). Nat Gas Industry, 36: 10–18
Korneev VA, Goloshubin G M, Daley T M, Silin D B. 2004. Seismic low-frequency effects in monitoring fluid-saturated reservoirs. Geophysics, 69: 522–532
Landau L D, Lifshitz E M. 1959. Elasticity Theory. Oxford: Pergamon Press
Landau L D, Lifshitz E M. 1987. Fluid Mechanics. 2nd ed. Oxford: Pergamon Press
Mase G T, Mase G E. 2001. Continuum Mechanics for Engineers. 2nd ed. Boca Raton: CRC Press
Mavko G, Mukerji T, Dvorkin J. 1998. Rock Physics Handbook: Tools for Seismic Interpretation in Porous Media. Cambridge: Cambridge University Press
Parra J O. 1997. The transversely isotropic poroelastic wave equation including the Biot and the squirt mechanisms: Theory and application. Geophysics, 62: 309–318
Pride S R, Gangi A F, Morgan F D. 1992. Deriving the equations of motion for porous isotropic media. J Acoust Soc Am, 92: 3278–3290
Quiroga-Goode G, Jiménez-Hernández S, Pérez-Flores M A, Padilla-Hernández R. 2005. Computational study of seismic waves in homogeneous dynamic-porosity media with thermal and fluid relaxation: Gauging Biot theory. J Geophys Res, 110: B07303
Sahay P N. 2001. Dynamic Green’s function for homogeneous and isotropic porous media. Geophys J Int, 147: 622–629
Sahay P N, Spanos T J, De la Cruz V. 2000. Macroscopic constitutive equations of an inhomogeneous and anistropic porous medium by volume averaging approach. SEG Technical Program Expanded Abstracts. 1834–1837
Sams M S, Neep J P, Worthington M H, King M S. 1997. The measurement of velocity dispersion and frequency-dependent intrinsic attenuation in sedimentary rocks. Geophysics, 62: 1456–1464
Schon J H. 2011. Physical Properties of Rocks. A workbook. Elsevier. 158
Spanos T J. 2001. The Thermophysics of Porous Media. Boca Raton: CRC Press
Spanos T J T. 2009. Seismic wave propagation in composite elastic media. Transp Porous Media, 79: 135–148
Spanos T J, Udey N, Dusseault M. 2002. Completing Biot Theory. In: Proceedings 2nd Biot Conf. on Poromechanics. 819–826
Wang D. 2016. Study on the rock physics model of gas reservoirs in tight sandstone (in Chinese). Chin J Geophys, 59: 4603–4622
Wang Z. 2013. Research progress, existing problem and development trend of tight rock oil (in Chinese). Petrol Geol Exper, 35: 7–15
Whitaker S. 1999. The Method of Volume Averaging: Theory and Applications of Transport in Porous Media. Dorderecht: Kluwer Academic
Winkler K W. 1985. Dispersion analysis of velocity and attenuation in Berea sandstone. J Geophys Res, 90: 6793–6800
Yang S, Liu W, Feng J, Wang R, Tu Z, Zhang Y, Tang Z, Hang D. 2008. Effect of pressure time on reservoir core permeability (in Chinese). J China Univ Petrol, 32: 0064–0068
Zhao H, Gao J, Liu F. 2014a. Frequency-dependent reflection coefficients in diffusive-viscous media. Geophysics, 79: T143–T155
Zhao H, Gao J, Zhao J. 2014b. Modeling the propagation of diffusive-viscous waves using flux-corrected transport-finite-difference method. IEEE J Sel Top Appl Earth Observ Remote Sens, 7: 838–844
Zhao X, Liao Q. 1983. Mechanics of Viscous Fluids (in Chinese). Beijing: China Machine Press
Zheng L, Liu J. 2019. Variation of seepage in one-dimensional low-permeable layer under low-frequency vibration. J Por Media, 22: 1519–1538
Zheng L, Zhang Y, Li Z, Ma P, Yang X. 2019. Rock consolidation seepage analysis under low frequency fluctuation considering different degree of porosity and pressure (in Chinese). Rock Soil Mech, 40: 1158–1196
Acknowledgements
The work was supported by the National Natural Science Foundation of China (Grant Nos. 41390450, 41390454, 91730306), the National Science and Technology Major Projects (Grant Nos. 2016ZX05024-001-007, 2017ZX05069), and the National Key R & D Program of the Ministry of Science and Technology of China (Grant No. 2018YFC0603501).
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Gao, J., Han, W., He, Y. et al. Seismic wave equations in tight oil/gas sandstone media. Sci. China Earth Sci. 64, 377–387 (2021). https://doi.org/10.1007/s11430-020-9686-0
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DOI: https://doi.org/10.1007/s11430-020-9686-0